B-Tree Insert and Delete Demo. Demo Demo slide by: Dr. J. Johnson.
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Transcript of B-Tree Insert and Delete Demo. Demo Demo slide by: Dr. J. Johnson.
B-Tree Insert and Delete Demo
Demo• Demo slide by: Dr. J. Johnson
• Suppose we start with an empty B-tree and keys arrive in the following order:1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
• We want to construct a B-tree of order 5• The first four items go into the root:
• To put the fifth item in the root would violate condition 4• Therefore, when 25 arrives, pick the middle key to make a
new root
Constructing a B-tree
1281 2
Constructing a B-treeAdd 25 to the tree
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
1281 2 25
Exceeds Order. Promote middle and split.
Constructing a B-tree (contd.)
6, 14, 28 get added to the leaf nodes:
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
12
8
1 2 25
12
8
1 2 2561 2 2814
Constructing a B-tree (contd.)
Adding 17 to the right leaf node would over-fill it, so we take the middle key, promote it (to the root) and split the leaf
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
12
8
2 2561 2 2814 2817
Constructing a B-tree (contd.)7, 52, 16, 48 get added to the leaf nodes
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
12
8
2561 2 2814
17
7 5216 48
Constructing a B-tree (contd.)
Adding 68 causes us to split the right most leaf, promoting 48 to the root
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
8 17
7621 161412 52482825 68
Constructing a B-tree (contd.)
Adding 3 causes us to split the left most leaf
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
48178
7621 161412 25 28 52 683 7
Constructing a B-tree (contd.)1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
Add 26, 29, 53, 55 then go into the leaves
481783
1 2 6 7 52 6825 28161412 26 29 53 55
Constructing a B-tree (contd.)
Add 45 increases the trees level
1 12 8 2 25 6 14 28 17 7 52 16 48 68 3 26 29 53 55 45
481783
29282625 685553521614126 71 2 45
Exceeds Order. Promote middle and split.
Exceeds Order. Promote middle and split.
Exercise in Inserting a B-Tree
• Insert the following keys to a 5-way B-tree:• 3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4,
31, 35, 56
Delete from a B-tree
• During insertion, the key always goes into a leaf. For deletion we wish to remove from a leaf. There are three possible ways we can do this:
• 1 - If the key is already in a leaf node, and removing it doesn’t cause that leaf node to have too few keys, then simply remove the key to be deleted.
• 2 - If the key is not in a leaf then it is guaranteed (by the nature of a B-tree) that its predecessor or successor will be in a leaf -- in this case can we delete the key and promote the predecessor or successor key to the non-leaf deleted key’s position.
Removal from a B-tree (2)
• If (1) or (2) lead to a leaf node containing less than the minimum number of keys then we have to look at the siblings immediately adjacent to the leaf in question: – 3: if one of them has more than the min’ number of keys then
we can promote one of its keys to the parent and take the parent key into our lacking leaf
– 4: if neither of them has more than the min’ number of keys then the lacking leaf and one of its neighbours can be combined with their shared parent (the opposite of promoting a key) and the new leaf will have the correct number of keys; if this step leave the parent with too few keys then we repeat the process up to the root itself, if required
Type #1: Simple leaf deletion
12 29 52
2 7 9 15 22 56 69 7231 43
Delete 2: Since there are enoughkeys in the node, just delete it
Assuming a 5-wayB-Tree, as before...
Note when printed: this slide is animated
Type #2: Simple non-leaf deletion
12 29 52
7 9 15 22 56 69 7231 43
Delete 52
Borrow the predecessoror (in this case) successor
56
Note when printed: this slide is animated
Type #4: Too few keys in node and its siblings
12 29 56
7 9 15 22 69 7231 43
Delete 72Too few keys!
Join back together
Note when printed: this slide is animated
Type #4: Too few keys in node and its siblings
12 29
7 9 15 22 695631 43
Note when printed: this slide is animated
Type #3: Enough siblings
12 29
7 9 15 22 695631 43
Delete 22
Demote root key andpromote leaf key
Note when printed: this slide is animated
Type #3: Enough siblings
12
297 9 15
31
695643
Note when printed: this slide is animated
Exercise in Removal from a B-Tree
• Given 5-way B-tree created by these data (last exercise):
• 3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56
• Add these further keys: 2, 6,12
• Delete these keys: 4, 5, 7, 3, 14